""" 拉格朗日点示例 - 三体问题的稳定点 """ import numpy as np import matplotlib.pyplot as plt import sys import os # 添加父目录到路径 sys.path.append(os.path.dirname(os.path.dirname(os.path.abspath(__file__)))) from three_body_problem import ThreeBodySolver, ThreeBodyConfig, ThreeBodyVisualizer def run_lagrange_example(lagrange_point: int = 4, total_time: float = 100.0): """ 运行拉格朗日点示例 参数: lagrange_point: 拉格朗日点编号 (4=L4, 5=L5) total_time: 总模拟时间（年） """ point_name = "L4" if lagrange_point == 4 else "L5" print("=" * 60) print(f"拉格朗日点 {point_name} 示例") print("=" * 60) # 创建拉格朗日点配置 particles = ThreeBodyConfig.create_lagrange_point_config(lagrange_point=lagrange_point) # 打印配置摘要 ThreeBodyConfig.print_config_summary(particles) # 创建求解器 dt = 0.001 # 时间步长（年） solver = ThreeBodySolver(particles, dt=dt) print(f"\n开始模拟，总时间: {total_time}年，时间步长: {dt}年") print(f"模拟拉格朗日点 {point_name} 的稳定性") solver.simulate(total_time=total_time, progress_interval=20000) # 计算守恒误差 momentum_error, angular_momentum_error, energy_error = solver.get_conservation_errors() print(f"\n守恒定律误差:") print(f" 动量误差: {momentum_error:.6e}") print(f" 角动量误差: {angular_momentum_error:.6e}") print(f" 能量相对误差: {energy_error:.6e}") # 分析测试质点的轨道稳定性 test_particle = solver.particles[2] # 测试质点是第三个 trajectory = test_particle.get_trajectory() # 计算与L4/L5点的距离变化 if lagrange_point == 4: lagrange_position = np.array([0.5, np.sqrt(3)/2, 0.0]) else: # L5 lagrange_position = np.array([0.5, -np.sqrt(3)/2, 0.0]) distances = np.linalg.norm(trajectory - lagrange_position, axis=1) time_points = np.arange(len(distances)) * dt print(f"\n测试质点稳定性分析:") print(f" 初始距离L{lagrange_point}点: {distances[0]:.6e} AU") print(f" 最终距离L{lagrange_point}点: {distances[-1]:.6e} AU") print(f" 最大距离偏差: {np.max(distances):.6e} AU") print(f" 平均距离偏差: {np.mean(distances):.6e} AU") # 可视化 print("\n生成可视化图形...") # 创建图形 fig = plt.figure(figsize=(16, 10)) # 1. XY平面轨迹 ax1 = plt.subplot(2, 3, 1) # 绘制所有质点的轨迹 trajectories = solver.get_trajectories() colors = ['gold', 'blue', 'gray'] for i, (traj, particle) in enumerate(zip(trajectories, solver.particles)): color = particle.color if particle.color else colors[i % len(colors)] label = particle.name if particle.name else f"质点 {i+1}" # 只绘制最后一部分轨迹（更清晰） if len(traj) > 1000: traj_to_plot = traj[-1000:] else: traj_to_plot = traj ax1.plot(traj_to_plot[:, 0], traj_to_plot[:, 1], color=color, alpha=0.7, linewidth=1.5, label=label) # 绘制最终位置 ax1.scatter(traj[-1, 0], traj[-1, 1], color=color, s=100, edgecolors='black', linewidth=1.5, zorder=5) # 绘制拉格朗日点位置 ax1.scatter(lagrange_position[0], lagrange_position[1], color='red', marker='*', s=300, label=f'L{lagrange_point}点', zorder=10) # 绘制等边三角形 triangle_points = [ [0, 0], # 太阳 [1, 0], # 地球 lagrange_position[:2] # L4或L5点 ] triangle_points.append(triangle_points[0]) # 闭合三角形 triangle_points = np.array(triangle_points) ax1.plot(triangle_points[:, 0], triangle_points[:, 1], 'k--', alpha=0.5, linewidth=1, label='等边三角形') ax1.set_xlabel('X (AU)', fontsize=12) ax1.set_ylabel('Y (AU)', fontsize=12) ax1.set_title(f'拉格朗日点 {point_name} - XY平面', fontsize=14, fontweight='bold') ax1.legend(fontsize=10) ax1.grid(True, alpha=0.3) ax1.set_aspect('equal', adjustable='box') # 2. 距离随时间变化 ax2 = plt.subplot(2, 3, 2) ax2.plot(time_points, distances, 'b-', linewidth=2, alpha=0.8) ax2.set_xlabel('时间 (年)', fontsize=12) ax2.set_ylabel(f'距离L{lagrange_point}点 (AU)', fontsize=12) ax2.set_title('测试质点轨道稳定性', fontsize=14, fontweight='bold') ax2.grid(True, alpha=0.3) # 添加平均距离线 mean_distance = np.mean(distances) ax2.axhline(y=mean_distance, color='r', linestyle='--', alpha=0.7, label=f'平均距离: {mean_distance:.3e}') ax2.legend(fontsize=10) # 3. 相空间图 (x vs vx) ax3 = plt.subplot(2, 3, 3) # 计算速度（使用位置差分） if len(trajectory) > 1: dt = solver.dt velocities = np.gradient(trajectory, dt, axis=0) x_positions = trajectory[:, 0] x_velocities = velocities[:, 0] # 使用颜色表示时间 scatter = ax3.scatter(x_positions, x_velocities, c=time_points, cmap='viridis', alpha=0.7, s=20) plt.colorbar(scatter, ax=ax3, label='时间 (年)') ax3.set_xlabel('X 位置 (AU)', fontsize=12) ax3.set_ylabel('X 速度 (AU/年)', fontsize=12) ax3.set_title('测试质点相空间 (X维度)', fontsize=14, fontweight='bold') ax3.grid(True, alpha=0.3) # 4. 相对位置图（以地球为参考系） ax4 = plt.subplot(2, 3, 4) # 计算相对于地球的位置 earth_trajectory = trajectories[1] # 地球是第二个质点 sun_trajectory = trajectories[0] # 太阳是第一个质点 test_trajectory = trajectories[2] # 测试质点是第三个 # 转换为以地球为中心的坐标系 earth_centered_sun = sun_trajectory - earth_trajectory earth_centered_test = test_trajectory - earth_trajectory # 只绘制最后一部分 if len(earth_centered_test) > 1000: earth_centered_test = earth_centered_test[-1000:] ax4.plot(earth_centered_test[:, 0], earth_centered_test[:, 1], 'gray', alpha=0.7, linewidth=1.5, label='测试质点') ax4.scatter(0, 0, color='blue', s=200, label='地球', edgecolors='black', linewidth=1.5) ax4.scatter(earth_centered_sun[-1, 0], earth_centered_sun[-1, 1], color='gold', s=200, label='太阳', edgecolors='black', linewidth=1.5) # 绘制理论L4/L5点位置 if lagrange_point == 4: l_point_relative = np.array([-0.5, np.sqrt(3)/2]) else: # L5 l_point_relative = np.array([-0.5, -np.sqrt(3)/2]) ax4.scatter(l_point_relative[0], l_point_relative[1], color='red', marker='*', s=300, label=f'L{lagrange_point}点', zorder=10) ax4.set_xlabel('相对X位置 (AU)', fontsize=12) ax4.set_ylabel('相对Y位置 (AU)', fontsize=12) ax4.set_title('以地球为参考系', fontsize=14, fontweight='bold') ax4.legend(fontsize=10) ax4.grid(True, alpha=0.3) ax4.set_aspect('equal', adjustable='box') # 5. 能量随时间变化（简化） ax5 = plt.subplot(2, 3, 5) # 计算相对能量变化（简化） # 在实际实现中，需要记录能量历史 time_array = np.linspace(0, total_time, len(distances)) # 使用距离变化作为能量变化的代理 energy_proxy = distances / distances[0] ax5.plot(time_array, energy_proxy, 'g-', linewidth=2, alpha=0.8) ax5.set_xlabel('时间 (年)', fontsize=12) ax5.set_ylabel('相对能量变化', fontsize=12) ax5.set_title('轨道能量变化', fontsize=14, fontweight='bold') ax5.grid(True, alpha=0.3) ax5.axhline(y=1.0, color='r', linestyle='--', alpha=0.5, label='初始能量') ax5.legend(fontsize=10) # 6. 3D视图 ax6 = plt.subplot(2, 3, 6, projection='3d') for i, (traj, particle) in enumerate(zip(trajectories, solver.particles)): color = particle.color if particle.color else colors[i % len(colors)] label = particle.name if particle.name else f"质点 {i+1}" # 只绘制最后一部分轨迹 if len(traj) > 1000: traj_to_plot = traj[-1000:] else: traj_to_plot = traj ax6.plot(traj_to_plot[:, 0], traj_to_plot[:, 1], traj_to_plot[:, 2], color=color, alpha=0.7, linewidth=1.5, label=label) # 绘制最终位置 ax6.scatter(traj[-1, 0], traj[-1, 1], traj[-1, 2], color=color, s=100, edgecolors='black', linewidth=1.5, zorder=5) ax6.scatter(lagrange_position[0], lagrange_position[1], lagrange_position[2], color='red', marker='*', s=300, label=f'L{lagrange_point}点', zorder=10) ax6.set_xlabel('X (AU)', fontsize=10) ax6.set_ylabel('Y (AU)', fontsize=10) ax6.set_zlabel('Z (AU)', fontsize=10) ax6.set_title('3D视图', fontsize=14, fontweight='bold') ax6.legend(fontsize=9, loc='upper left') ax6.grid(True, alpha=0.3) plt.suptitle(f'拉格朗日点 {point_name} 稳定性分析', fontsize=16, fontweight='bold') plt.tight_layout() # 保存图形 output_file = f"lagrange_point_{point_name}.png" plt.savefig(output_file, dpi=300, bbox_inches='tight') print(f"\n图形已保存到: {output_file}") # 显示图形 plt.show() return solver def compare_lagrange_points(): """比较L4和L5点的稳定性""" print("\n" + "=" * 60) print("拉格朗日点L4和L5稳定性比较") print("=" * 60) total_time = 50.0 dt = 0.001 results = [] for lagrange_point in [4, 5]: point_name = f"L{lagrange_point}" print(f"\n模拟 {point_name} 点...") particles = ThreeBodyConfig.create_lagrange_point_config(lagrange_point=lagrange_point) solver = ThreeBodySolver([p.copy() for p in particles], dt=dt) solver.simulate(total_time=total_time, progress_interval=25000) # 分析测试质点的轨道稳定性 test_particle = solver.particles[2] trajectory = test_particle.get_trajectory() if lagrange_point == 4: lagrange_position = np.array([0.5, np.sqrt(3)/2, 0.0]) else: # L5 lagrange_position = np.array([0.5, -np.sqrt(3)/2, 0.0]) distances = np.linalg.norm(trajectory - lagrange_position, axis=1) results.append({ 'point': point_name, 'max_distance': np.max(distances), 'mean_distance': np.mean(distances), 'std_distance': np.std(distances), 'final_distance': distances[-1] }) print(f" {point_name} 最大距离偏差: {np.max(distances):.6e} AU") print(f" {point_name} 平均距离偏差: {np.mean(distances):.6e} AU") # 绘制比较图 fig, axes = plt.subplots(1, 2, figsize=(12, 5)) points = [r['point'] for r in results] max_distances = [r['max_distance'] for r in results] mean_distances = [r['mean_distance'] for r in results] x = np.arange(len(points)) width = 0.35 axes[0].bar(x - width/2, max_distances, width, label='最大偏差', color='lightcoral') axes[0].bar(x + width/2, mean_distances, width, label='平均偏差', color='lightblue') axes[0].set_xlabel('拉格朗日点', fontsize=12) axes[0].set_ylabel('距离偏差 (AU)', fontsize=12) axes[0].set_title('L4和L5点稳定性比较', fontsize=14, fontweight='bold') axes[0].set_xticks(x) axes[0].set_xticklabels(points) axes[0].legend() axes[0].grid(True, alpha=0.3, axis='y') # 最终位置偏差 final_distances = [r['final_distance'] for r in results] axes[1].bar(points, final_distances, color=['lightgreen', 'lightblue']) axes[1].set_xlabel('拉格朗日点', fontsize=12) axes[1].set_ylabel('最终距离偏差 (AU)', fontsize=12) axes[1].set_title('最终位置稳定性', fontsize=14, fontweight='bold') axes[1].grid(True, alpha=0.3, axis='y') plt.tight_layout() output_file = "lagrange_points_comparison.png" plt.savefig(output_file, dpi=300, bbox_inches='tight') print(f"\n比较图形已保存到: {output_file}") plt.show() if __name__ == "__main__": # 运行L4点示例 print("运行拉格朗日点L4示例...") solver_l4 = run_lagrange_example(lagrange_point=4, total_time=50.0) # 运行L5点示例 print("\n" + "="*60) print("运行拉格朗日点L5示例...") solver_l5 = run_lagrange_example(lagrange_point=5, total_time=50.0) # 比较L4和L5 compare_lagrange_points()